Kroos had the ball in the net with just four minutes gone at the Alfredo Di Stefano stadium, where Madrid will play while the Bernabeu is redeveloped. Ramos got a second with half an hour gone after Karim Benzema and Eden Hazard combined, before Marcelo made it 3-0.
Madrid needed a winning start and a performance to match Barca's comprehensive victory in Mallorca and got both, especially in the first half.
Coach Zinedine Zidane insisted his team would have no problems adapting to the Di Stefano and so it proved. There was a seriousness about Madrid's play from the start, while the quality of the pitch looked every bit as good as that of the Bernabeu.
The other big plus was Hazard. Three months after ankle surgery that should have ended his season, he started, played an hour, grabbed an assist and looked as sharp as at any point since joining the club last summer.
Dani Carvajal was taken off at the interval, while Ramos and Marcelo both looked to be struggling with knocks. Zinedine Zidane likes to rotate and made all five subs available to him; Madrid play Valencia in just four days and the club's large squad might be tested to its limit over the next month.
Meanwhile, though the three points never looked in danger given the 3-0 interval lead, the second half was more of a challenge against an improved Eibar side, with Courtois called into action to make multiple saves and conceding a poor goal.
Manager rating out of 10
8 -- It was Zidane's 200th game in charge of Real Madrid and one that he had admitted would be the strangest. The first-half display justified a somewhat unexpected team selection, with Rodrygo in attack and a three-man midfield; the trio of Casemiro, Kroos and Modric had often been lacking earlier this season but proved effective here.
Player ratings (1-10; 10 = best. Players introduced after 70 minutes get no rating)
GK Thibaut Courtois, 7 -- A quiet first half was followed by a much busier second period, in which Courtois had to make some smart saves. He will not be happy with Eibar's goal squirming past him, though.
DF Dani Carvajal, 7 -- Taken off after 45 minutes for what Madrid will be hoping was just a precaution. The right-back spot is a worry with Nacho also out injured.
DF Sergio Ramos, 8 -- The captain has returned from lockdown looking in better shape than ever. He started and finished the move that made it 2-0 with a trademark burst forward.
DF Raphael Varane, 7 -- The French international made one vital interception when the score was 1-0 and had to lead the defence when Ramos was taken off with half an hour to go.
DF Marcelo, 8 -- Picked ahead of Ferland Mendy, although both will get plenty of game time this month, Marcelo had to deal with Eibar's talented Orellana. He marked his first league goal since October 2018 by taking a knee.
MF Casemiro, 7 -- Picked up a yellow card for a clumsy challenge early in the second half, but had a tidy game otherwise.
MF Luka Modric, 7 -- Started well and looked refreshed before fading and being replaced. He remains an important presence despite only starting just over half of Madrid's league games this season.
MF Toni Kroos, 8 -- Scored superbly to give Madrid a dream start. Kroos had a difficult time in February and March, when he was benched against Manchester City and subbed in a defeat to Real Betis, but this was a big improvement.
FW Rodrygo, 8 -- A surprise pick on the right wing ahead of Bale or Marco Asensio, Rodrygo went close to scoring just before half-time but was largely quiet.
FW Eden Hazard, 8 -- Back on the pitch three months after ankle surgery, Hazard played like he had a point to prove in his 60 minutes. Unselfishly, he laid on a pass to Ramos for the second goal.
FW Karim Benzema, 6 -- More important than ever with Luka Jovic and Mariano Diaz absent. Benzema's vision spotted Hazard in space for the game-clinching second goal.
DF Ferland Mendy, 6 -- Replaced Carvajal at half-time and had to play out of position on the right. Made one especially good run late on.
FW Gareth Bale, 6 -- Reportedly beat all-comers in post-lockdown fitness tests, but did not make the most of his half-hour run-out.
FW Vinicius Junior, 6 -- Replaced Rodrygo and looked to trouble Eibar with his pace in behind.
DF Eder Militao, 6 -- Came on for Ramos as Madrid looked to consolidate their advantage.
MF Federico Valverde, NR -- Played the last few minutes in place of Modric.